# Inverse parabolic partial differential equation

An equation of the form

$$\tag{* } u _ {t} + \sum _ {i , j = 1 } ^ { n } a _ {ij} ( x , t ) u _ {x _ {i} x _ {j} } - \sum _ { i= } 1 ^ { n } a _ {i} ( x , t ) u _ {x _ {i} } - a ( x , t ) u =$$

$$= \ f ( x , t ) ,$$

where the form $\sum a _ {ij} \xi _ {i} \xi _ {j}$ is positive definite. The variable $t$ plays the role of "inverse" time. The substitution $t = - t ^ \prime$ reduces equation (*) to the usual parabolic form. Parabolic equations of "mixed" type occur, for example, $u _ {t} = x u _ {xx}$ is a direct parabolic equation for $x > 0$ and an inverse parabolic equation for $x < 0$, with degeneracy of the order for $x = 0$.

$$u _ {t} + \Delta u = 0$$
( $\Delta$ being the Laplace operator) see [a1].